Optimal Interpolation of Structural

2 This paper presents a method for interpolating a sparse set of velocity measurements on the surface of a vibrating structure. The method is based on a novel integration of a physical model of the vibrating structure with an optimal signal recovery technique. The ability to achieve good interpolation performance with a small set of sensors makes this method important for many active noise control applications where an accurate measure of structural surface velocity is required to predict the radiated acoustic eld. To demonstrate the applicability of this method in active control applications, a sample experiment is performed and results presented. 1 The interior of the ellipse represents the space of all possible signals b = Ax. If one sample of the two-dimensional signal vector b is known (b 1), then the other sample b 2 is restricted to lie on the intersection of the ellipse interior and the vertical dashed line. The optimal solution in this case is b opt , which is the midpoint between points b max 2 and b min 2 Plate mode shapes within the 480-510 Hz frequency range: (a) 481.8 Hz (b) 483.3 Hz (c) 503.6 Hz (d) 505.8 Hz 3 Plate velocity response at 495 Hz due to a 0.8 N force applied at x = 0:1m and y = 0:05m. : 14 4 Performance of interpolation technique as a function of the number of sensors. The sensor locations were chosen for uniform coverage of the plate. The normalized error was computed by 20 log(k v ? ^ v k = k v k 5 Plot of error j v? ^ v j for 9 sensors, 16 sensors, and 25 sensors out of the possible 209 locations. The vertical scale corresponds to the error which is displayed in the lower mesh plot. The * symbol within the rectangular box above each graph correspond to the sensor locations. : : : 16 6 Plot of interpolated normal velocity responses for 25 sensors out of the possible 209 locations, with (a) actual velocity response, (b) estimate using cubic spline interpolation, (c) estimate with bilinear interpolation, and (d) estimate with new model-based method. The * symbol within the rectangular box above each graph correspond to the sensor 7 Predicted reductions in overall radiated sound pressure from a rectangular plate (negative numbers = reductions, positive numbers = enhancements). The methodology presented in this paper (solid line) is …